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Home , Anthemius of Tralles, Autolycus of Pitane, Dinostratus, Hippocrates of Chios, History of arithmetic, Hypsicles

SCIENCE AWAKENING B.L.VAN DER WAERDEN SCIENCE AWAKENING

I

English translation by A rnold Dresden with additions of the author

Fourth edition

KLUWER ACADEMIC PUBLISHERS, DORDRECHT, THE NETHERLANDS SCHOLAR'S BOOKSHELF, PRINCETON JUNCTION, NEW JERSEY, U.S.A. Hardcover edition published throughout the world, exclusive of North America, by Kluwer Academic Publishers, Spuilboulevard 50, P.O. Box 17, 3300 AA Dordrecht, The Netherlands

Paperback edition published throughout the world by The Scholar's Bookshelf, 51 Everett Drive, P.O. Box 179, Princeton Junction, New Jersey 08550, United States of America

Hardcover edition published in North America by The Scholar's Bookshelf

Copyright © 1975 by Noordhoff International Softcover reprint ofthe hardcover 4th edition 1975 Publishing, a division of Kluwer Academic Publishers, Dordrecht, The Netherlands

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form by any means, electronic, mechanical, photocopying, recording or otherwise without the prior permission of the copyright owner.

First edition 1954 Second edition 1%1 Third edition 1969 Fourth edition 1975 Fifth edition 1988

Kluwer Academic Publishers

ISBN-13: 978-94-010-7115-4 ~ISBN-13: 978-94-009-1379-0 DOl: 10.1007/978-94-009-1379-0

First Scholar's Bookshelf hardcover printing, 1988 First Scholar's Bookshelf paperback printing, 1988 PREFACE TO THE ENGLISH EDITION

Soon after the publication of my"Ontwakende W etenschap"the need for an English translation was felt. We were very glad to find a translator fully familiar with the English and Dutch languages and with mathematical terminol· ogy. The publisher, Noordhoff, had the splendid idea to ask H. G. Beyen, professor of archeology, for his help in choosing a nice set of illustrations. It was a difficult task. The illustrations had to be both instructive and attractive, and they had t~ illustrate the history of science as well as the general background of ancient civilization. The publisher encouraged us to find better and still better illustrations, and he ordered photographs from all over the world, with never failing energy and enthusiasm. Mr. Beyen's highly instructive subscripts will help the reader to see the inter· relation between way of living, art, and science of the ancient world.

Thanks are due to many correspondents, who have suggested additions and pointed out errors. Sections on and Stereographte Projection and on ' construction of the heptagon have been added. The sections on Perspective and on the Anaphorai of have been enlarged.

In the second English edition I have incorporated an important discovery of P. Huber, which sheds new light upon the role of In Babylonian (see p. 73). The section on Heron's Metrics (see p. 277) was written anew, follOWing a suggestion of E. M. Bruins.

Zurich. 1961 B. L. VAN DER WAERDEN PREFACE

Why ?

Every one knows that we are living in a technological era. But it is not often realized that our technology is based entirely on mathematics and physics. When we ride home on the streetcar in the evening, when we turn on the electric light and the radio, everything depends on cleverly constructed physical mechanisms based on mathematical calculations. But more than that! We owe to physics not only these pleasant articles of luxury, but, to a large extent, even our daily bread. Apart from the fact that our grains come to us, chiefly by steamer from overseas, our own agriculture would be far less productive without artificial fertilizers. Such fertilizers are chemical products, and chemistry depends on physics. l But science has not brought us blessings only. The destructive armaments which mankind uses at the present time to knock its own civilization to pieces are also products to which the development of mathematics and physics have inevitably led. Our spiritual life is also influenced hy science and technology, in a measure but rarely fully understood. The unprecedented growth of natural science in the 17th century was followed ineluctably by the rationalism of the 18th, by the deification of reason and the decline of religion; an analogous development had taken place earlier, in Greek times. In a similar manner, the triumphs of technology in the 19th century were followed in the 20th by the deification of technology. Unfor• tunately, man seems to be overly inclined to deify whatever is powerful and successful. These considerations indicate that science has put its stamp on the whole of our life, material and spiritual, in its beneficent and in its evil aspects. Science is the most significant phenomenon of modern times, the principal ingredient of our civilization - alas! But if this be true, then the most important question for the history of culture is: How did our modern natural science come about? It will be conceded that most historical writings either do not consider this question at all, or else deal with it in a very unsatisfactory manner. For example, which are the histories of Greek culture that mention the names of and of Eudoxus, two of the greatest of all times? Who realizes that, from the historical point of view, Newton is the most important figure of the 17th century? Every physicist will admit that the mechanics of Newton are the foundation of modern physics. Every knows that modern astronomy begins with Kepler and Newton. And every knows that the largest domain of

1 The ruder should bur in mind that this book was addressed originally to the public of the N,·thcrl.nds. 4 PREFACE modern mathematics, the part most important for physics, is Analysis, which has its roots in the Differential and Integral Calculus of Newton. Thus the work of Newton constitutes the foundation for by far the greater part of modern exact science. It was Newton who discovered the fundamental laws of motion, to which terrestrial as well as celestial objects are subject. He placed the crown on the task of renovating antique astronomy, begun by Copernicus and Kepler. He discovered a general method for solving all problems of differentiation and integration, whereas Archimedes, the greatest genius of antiquity, had not progressed beyond special methods for particular problems. The work of Newton can not be understood without a knowledge of antique science. Newton did not create in a void. Without the stupendous work of Ptole• my, which completed and closed antique astronomy, Kepler's Astronomia Nova, and hence the mechanics of Newton, would have been impossible. Without the conic sectIOns of Apollonius, which Newton knew thoroughly, his deVelopment of the law of gravitation is equally unthinkable. And Newton's integral calculus can be understood only as a continuation of Archimedes' determination of and volumes. The history of mechanics as an exact science begins '.vith the laws of the lever, the laws of hydrostatics and the determinatIOn of mass centers by Archimedes. In short, all the developments which converge in the work of Newton,"1hose of mathematics, of mechanics and of astronomy, begin in .

The History of , from fhales to Apollonius, covers the four centuries from 600 B.C. to 200 B.C. Until recently, the first three of these four centuries were enveloped in twilight, because we possess only two original texts from this period: the fragment con• cerning the lunules of and that of on the duplication of the cube. To this can be added two brief fragments of Archytas, a of scattered communications of , , Pappus, and Eutocius, and a self• contradictory set of Pythagorean legends. For this reason, the older works, such a~ Cantor's Geschichte der Mathematik, contain little more about this period than speculations concerning things of which we really do not know anything, such as, for example, the "Theorem of " In recent times however more light has penetrated into the darkness. In the first place, as a result of the indefatigable industry of Otto Neugebauer and his collaborators, we know now the mathematical cuneiform texts, which have thrown an entirely new light not only on the Theorem of Pythagoras, but especially on the earliest history of and of algebra. Neugebauer, following in the tracks of Zeuthen, succeeded in discovering the hidden algebraic element in Greek mathematics and in demonstrating its connection with Babylonian algebra. No PREFACE 5 longer does the begin with ; it starts 2000 years earlier in Mesopotamia. And, as to arithmetic, in 1937 Neugebauer wrote: "What is called Pythagorean in the Greek tradition, had probably better be called Babylonian"; and, a cuneiform text, concerning "Pythagorean ", dis• covered in 1943, showed that he was entirely right. A second new impulse came from philosophically oriented philology. In 1927, Stenzel and Toeplitz, with Neugebauer, established the penodical "~ellen und Studien zur Geschichte der Mathematik, Astronomie und Physik". It was their purpose to get to know more about the philosophy of Plato by an analysis of the fundamental concepts of Greek mathematics, and, reciprocally, to learn more about Greek mathematics by means of an analysis of Plato. This method has enabled Becker, Reidemeister and others to obtain highly important results. At an earlier date, Eva Sachs had rescued the excellent mathematician Theaetetus from oblivion. Another very fertile method was the analysis of the Elements of . This work, written about 300 B.C., proves to be largely a compilation of mathematical fragments, quite diverse in calibre and quite varied in age. By carefully taking these fragments apart, by dusting them off and then replacing them in the mathe• matical historical environment from which they had originally come, it has become possible to obtain a considerably clearer picture of Greek mathematICS of the years 500---300 B.C. These different things have not as yet been brought together in a book. We have, it is true, an excellent book by O. Neugebauer on "Vorgriechische Mathe• matik", and there are excellent works on Euclid, Aristarchus, Archimedes and Apollonius. As to these latter writers, we shall therefore be able to confine our• selves to the most interesting and the most important things; we shall pick tidbits here and there from their works and we shall try to serve them in as tasteful a manner as possible. But the principal purpose of the present book is to explain clearly how Thales and Pythagoras took their start from Babylonian mathematics but gave it a very different, a specifically Greek character; how, in the Pythagorean school and outside, mathematics was brought to higher and ever higher development and began gradually to satisfy the demands of stricter logic; how, through the work of Plato's friends Theaetetus and Eudoxus, mathematics was brought to the state of perfection, beauty and exactness, which we admire in the elements of Euclid. We shall see moreover that the mathematical method of proof served as a prototype for Plato's dialectiCS and for Aristotle's logic.

The history of mathematics should not be detached from the general history of culture. Mathematics is a domain of intellectual activity, intimately related not only to astronomy and mechanics, but also to architecture and technology, to philosophy, and even to religion (Pythagoras!). 6 PREFACE

Political and social conditions are of very great importance for the flowering of science and for its character. This will become very clear in Chapter VII when the mathematics of is compared with that of the classical period, and still more in Chapter VIII in the discussion of the causes for the decay of Greek mathematics. The plan of this book. It is the intention to make this book scientific, but at the same time accessible to anyone who has learned some mathematics in school and in college, and who is interested in the history of mathematics. It is to be scientific in the sense that it is to be based on a study of the sources and that its conclusions are to be supported by arguments, so as to enable the reader to judge the conclusions for himself. The naive reader may take the use of such a method for granted. But - how often has it been sinned against! How frequently it happens that books on the history of mathematics copy their assertions uncritically from other books, without consulting the sources! How many fairy tales circulate as "universally known truths" ! Let us quote an example. In 90 %of all the books, one finds the statement that the Egyptians knew the of sides 3, 4 and 5, and that they used it for laying out right . How much value has this statement? None! What is it based on! On two facts and an argument of Cantor. The facts are the following: "rope-stretchers" took part in laying out an Egyptian temple, and the angles at the base of temples and pyramids are nearly always, very accurately, right angles. Now Cantor reasons as follows: these right angles must have been constructed by the rope-stretchers, and I (Cantor) can not think of any other way of constructing a right by means of stretched ropes than by using three ropes of lengths 3, 4 and 5, forming a right triangle. Therefore the Egyptians must have known this triangle. Is this not incredible? Not that Cantor at one time formulated this hypothesis, but that repeated copying made it a "universally known fact". This is nevertheless the fact. To avoid such errors, I have checked all the conclusions which I found in mo• dern writers. This is not as difficult as might appear, even if, as is my case, one cannot read either the Egyptian characters or the cuneiform symbols, and one is not a classical philologist. For reliable translations are obtainable of nearly all texts. For example, Neugebauer has translated and published all mathematical cuneiform texts. The Egyptian mathematical texts have all been translated into English or German. Plato, Euclid, Archimedes, ... of all these, good translations exist in French, German and English. Only in a few doubtful cases it became necessary to consult the Greek text. Not only is it more instructive to read the classical authors themselves (in translation if necessary), rather than modern digests, it also gives much greater PREFACE 7 enjoyment. If my book should lead the reader to do this, it will fully have accom• plished its purpose. For this reason, I advise the reader emphatically not to accept anything on my say-so, but to verify everything.

I have tried to consider the great mathematicians as human beings living in their own environment and to reproduce the impression which they made on their contemporaries. In some cases, the scarcity of source material made this impossible. but striking personalities such as Pythagoras, Archytas, Theaetetus ana Archimedes can be made to stand out clearly. It is also possible to get an impression of the character of Thales, Eudoxus and . Of the Egyptian and Babylonian mathematicians not even the names are known.

What is new in this book, In Chapter II. A hypothesis of Freudenthal on Indian number symbols. In Chapter III. Freudenthal's interpretation of a Babylonian textbook. In Chapter IV. A new way of looking at the mathematics of Thales. In Chapter V. Reconstruction of the Pythagorean theory of numbers from the arithmetical books of the Elements. Connections between the Babylonian and the Greek mathematics, particularly the Pythagorean mathematics. The irrationality proofs of . In Chapter VI. The feeble logic of Archytas of Taras. Mathematics and the theory of harmony in the Epinomis. An analysis of Book X of the Elements and a reconstruction of the mathematical work of Theaetetus. In Chapter VII. The history of the Delian problem, actually and according to the dialogue Platonicus. In Chapter VIII. The cause of the decay of Greek mathematics.

Acknowledgements. I thank Dr. Brinkman, Professor Freudenthal and especially Dr. Dijksterhuis, who have read the manuscript critically and have made numerous useful remarks. 8 PREFACE

I thank the many others who have helped with brief observations or with tech• nical advice. I am very much obliged to Mr. Wijdenes who has taken care of the diagrams in his well-known careful manner, assisted by his excellent draughtsman, Mr. Bousche. In conclusion, I thank the publishers for the generous way in which they have taken all my wishes into account. TABLE OF CONTENTS

PREFACE ...... '>-8 Why History of Mathematics. . . . 3 The History of Greek. Mathematics. 4 The plan of this book . . 6 What is new in this book . 7 Acknowledgements. . . . . 7 CHAPTER I. The Egyptians 15-36 Chronological Summary. . . 15 The Egyptians as the "inventors" of geometry . 15 The Rhind papyrus ...... 16 For whom was the Rhind papyrus written? 16 The \..lass of royal scribes. . . 17 The technique of calculation ...... 17 Multiplication...... 18 Division ...... 19 Natural fractions and unit fractions. 19 Calculation with natural fractions. . 20 Further relations between fractions . 21 Duplication of unit fractions 22 Division once more. 23 The (2 : n) table ...... 23 The red auxiliaries . . . . . 26 Complementation of a fraction to 1. 26 "Aha·calculations" ...... 27 Applied calculations ...... 29 The development of the computing technique 30 Hypothesis of an advanced science ...... 31 The geometry of the Egyptians. 31 Inclination of oblique planes 31 Areas ...... 32 of the hemisphere. 33 Volumes ...... 34 What could the learn from the Egyptians? 35 CHAPTER II. Number systems, digits and the art of computing. 37-61 The system...... 37 How did the sexagesimaI system originate? . 40 Oldest Sumerian period (before 3000 B.C.) 40 Later Sumerian period (about 2000 B.C.) . 40 Sumerian technique of computation. . . . 42 Table of 7 and of 16.40 ...... 42 Normal table of inverses ...... 43 Squares. square roots and cube roots. 44 The Greek notation for numbers...... 45 Counting boards and counting pebbles. 47 Calculation with fractions...... 48 10 TABLE OF CONTENTS

Sexagesimal fractions ...... 50 Hindu numerals...... 51 Number systems; Kharosti and Brahmi .. 53 The invention of the positional system. 53 The date of the invention ...... 53 Poetic numbers ...... 54 Aryabhata and his syllable-numbers .. 55 Where does the zero come from? .. 56 The triumphal procession of the Hindu numerals . 57 The abacus of Gerbert...... 58 CHAPTER III. Babylonian mathematics __ . 62-81 Chronological summary...... 62 Babylonian algebra ...... 63 First example (MKT I. p. 113). . 63 Interpretation...... 63 Second example (MKT I. p. 280). 65 Third example (MKT I. p. 323) . 66 Fourth example (MKT I. p. 154). 68 Fifth example (MKT III. p. 8. no. 14) . 68 Quadratic equations (MKT III. p. 6). . 69 Sixth example (MKT III. p. 9. no. 18). 70 Seventh example (MKT I. p. 485) . . . 70 Eighth example (MKT I. p. 204). . . . 71 Geometrical proofs of algebraic formulas? 71 Ninth example (MKT I. p. 342) 72 A lesson-text (MKT II. p. 39) ...... 73 Babylonian geometry...... 75 Volumes and areas ..... _ ...... 75 Frustra of and of pyramids (MKT. pp. 176 and 178) . 75 The "Theorem of Pythagoras" (MKT II. p. 53) ...... 76 Babylonian theory of numbers ...... 77 Progressions (MKT I, p. 99) _ ...... 77 Plimpton 322: Right triangles with rational sides. 78 Applied mathematics . . . . _ _ ...... 80 Summary ...... _ .. . 80 GREEK MATHEMATICS CHAPTER IV. The age of Thales and Pythagoras. . . 82-104 Chronological summary...... 82 Hellas and the Orient ...... 83 Thales of Milete ...... 85 Prediction of a . . 86 The geometry of Thales . 87 From Thales to Euclid. . . . 90 Pythagoras of . . . . . 92 The travels of Pythagoras 94 Pythagoras and the theory of harmony. 95 Pythagoras and the theory of numbers . 96 Perfect numbers ...... 97 TABLE OF CONTENTS 11

Amicable numbers . . . . 98 Figurate numbers. . . . . 98 Pythagoras and geometry. 100 The astronomy of the Pythagoreans 101 Summary ...... 102 The tunnel on Samos...... 102 Antique measuring instruments. 104 CHAPTER V. The golden age .. 106-147 ...... 106 The Mathemata of the Pythagoreans . 108 The theory of numbers...... 108 The theory of the even and the odd. 108 Proportions of numbers ...... 110 The solution of systems of equations of the first 116 geometry: ....." ...... 116 Geometnc Algebra ...... 118 Why the geometric formulation? . 125 Lateral and diagonal numbers. . 126 of Clazomenae . 127 of Abdera . . . 128 of . . . . . 129 Squaring the . . . . . 130 Antiphon ...... 130 . . . . 131 in the fifth century. and Perspective. 136 Demo.::ritus...... 137 and pyramid . . . . . 137 Plato on solid geometry . . 138 The duplication of the cube 139 Theodorus of Cyrene. . . . 141 Theodorus and Theaetetus . 1-41 Theodorus on higher curves and on mixtures. 146 and his ...... 1-46 The main lines of development. . . . 146 CHAPTER VI. The century of Plato. 14&-200 Archytas of Taras ...... 149 Tht: duplication of the cube . . . 150 The style of Archytas ...... 152 Book VIII of the Elements. . . . 153 The Mathemata in the Epinomis . 155 The duplication of the cube . . . . . 159 According to . . . . 162 Theaetetus ...... 165 Analysis of Book X of the Elements . 168 The theory of the regular polyhedra . 173 The theory of proportions in Theaetetus . 175 Eudoxus of Cnidos...... 179 Eudoxus as an astronomer ...... 180 12 TABLE OF CONTENTS

The exhaustion method. . . 184 The theory of proportions . 187 Theaetetus and Eudoxus . 189 Menaechmus .... 190 ...... 191 . . . . . 193 On the rotating . 194 On the rising and setting of stars 195 Euclid ...... 195 The "Elements" 196 The "" ... 198 On the division of ligures. 199 Lost geometrical writings. 200 Euclid's work on applied mathematics 200 CHAPTER VII. The Alexandrian Era (330-200 B.C.). 201-263 ...... 202 Archimedes' measurement of the circle . 204 Tables for the lengths of chords . 206 Archimedes...... 208 Stories about Archimedes. . . 208 Archimedes as an astronomer . 211 The works of Archimedes . . 211 The "Method" ...... 212 The quadrature of the . 216 On sphere and cylinder I. 220 On sphere and cylinder II . . . 222 ...... 223 . . . 223 The notion of integral in Archimedes. 224 The ...... 225 The construction of the regular heptagon. 226 The other works of Archimedes 227 Eratosthenes of Cyrene ...... 228 Life ...... 228 Chronography and measurement of a degree . 230 Duplication of the cube. 2.30 Theory of numbers. . . . . 231 Medieties ...... 231 Nicomedes ...... 235 The trisection of the angle . 236 The duplication of the cube in Nicomedes 236 ...... 237 The theory of the epicycle and of the excenter. 238 Conica ...... 240 The conic sections before Apollonius...... 241 The as a sfction of a cone according to Archimedes 243 How were the symptoms derived originally? ...... 245 A question and an answer ...... 245 The derivation of the symptoms according to Apollonius. 246 and conjugate . . . 248 TABLE OF CONTENTS 13

Tangent lines...... 249 The equation referred to the center ...... 251 The two-tangents theorem and the transformation to new axes. 252 Cones of revolution through a given conic . 256 The second book...... 258 The third book...... 258 Loci involving 3 or 4 straight lines. 259 The fifth book ...... 260 The sixth. seventh and eighth books . 261 Further works of Apollonius. . . 261 CHAPTER VIII. The decay of Greek mathematics. . 264-291 External causes of decay ...... 264 The inner causes of decay ...... 265 1. The difliculty of geometric algebra. . . . 265 2. The difliculty of the written tradition . . 266 The commentaries of Pappus . 267 The epigones of the great mathematicians. . 267 1. Diodes .... 267 The cissoid. . . . . 268 2. Zenodorus...... 268 Isoperimetric figures. 269 3. Hypsicles ...... 269 The fourteenth book of the Elements. 269 Anaphora ...... 270 . . 271 Plane trigonometry. . . 271 Spherical trigonometry . 274 Menelaus ...... 274 Transversal proposition. 275 Heron of Alexandria . . . 276 Metrics ...... 277 Diophantus of Alexandria. 278 ...... 278 Diophantine equations 279 The precursors of Diophantus 279 Connection with Babylonian and Arabic algebra . 280 The algebraic symbolism 281 From Book II . 282 From Book III . 283 From Book IV . 284 From Book V . 285 From Book VI . 285 . 286 A of Euclid. 287 The theorem on the . 288 Theorem of Pappus...... 289 ...... _ . 290 ...... " ...... 290 The school. Proclus Diadochus. . . 290 Isidore of Milete and . 291